The only difference between this case and the case of permutations without
replacement is that the selection order of the k selected objects is
not distinguished here. This implies that a different arrangement of the same
objects is not counted for this case and so this case involves simply selecting
a subset of size k from the population. Therefore, we can view the
number of permutations without replacement as a two-stage process: first select
a subset (combinations without replacement) and then generate every possible
rearrangement of each of these subsets. Note that the number of ways to
generate every possible rearrangement of k objects is equivalent to
counting the number of permutations without replacement of k objects
selected from a population of size k, and so is equal to k!.
Denote by C(n,k) the number of combinations without replacement. Then
we have,